Quadratic Sequences
What is a quadratic sequence?
- Unlike in a linear sequence, in a quadratic sequence the differences between the terms (the first differences) are not constant
- However, the differences between the differences (the second differences) are constant
- Another way to think about this is that in a quadratic sequence, the sequence of first differences is a linear sequence
eg Sequence: 2, 3, 6, 11, 18, …
1st Differences: 1 3 5 7 (a Linear Sequence)2nd Differences: 2 2 2 (Constant)
- If the second differences there are constant, we know that the example is a quadratic sequence
What should we be able to do with quadratic sequences?
- You should be able to recognise and continue a quadratic sequence
- You should also be able to find a formula for the nth term of a quadratic sequence in terms of n
- This formula will be in the form:
nth term = an2 + bn + c
(The process for finding a, b, and c is given below)
How do I find the nth term of a quadratic sequence?
- Work out the sequences of first and second differences
Note: check that the first differences are not constant and the second differences are constant, to make sure you have a quadratic sequence!
- e.g. sequence: 1, 10, 23, 40, 61
- first difference: 9, 13, 17, 21, ...
- second differences: 4, 4, 4, ...
- e.g. sequence: 1, 10, 23, 40, 61
- a = [the second difference] ÷ 2
- e.g. a = 4 ÷ 2 = 2
- e.g. a = 4 ÷ 2 = 2
- Write out the first three or four terms of an2 with the first three or four terms of the given sequence underneath.
Work out the difference between each term of an2 and the corresponding term of the given sequence.- e.g. an2 = 2n2 = 2, 8, 18, 32, ...
sequence = 1, 10, 23, 40, ...
difference = -1, 2, 5, 8, ...
- e.g. an2 = 2n2 = 2, 8, 18, 32, ...
- Work out the linear nth term of these differences. This is bn + c.
- e.g. bn + c = 3n − 4
- e.g. bn + c = 3n − 4
- Add this linear nth term to an2. Now you have an2 + bn + c.
- e.g. an2 + bn + c = 2n2 + 3n − 4
Examiner Tip
- Before doing the very formal process to find the nth term, try comparing the sequence to the square numbers 1, 4, 9, 16, 25, … and see if you can spot the formula
- For example:
- Sequence 4, 7, 12, 19, 28, …
- Square Numbers 1 4 9 16 25
- We can see that each term of the sequence is 3 more than the equivalent square number so the formula is
nth term = n2 + 3 - This could save you a lot of time!
Worked example
For the sequence 5, 7, 11, 17, 25, ....
(a)
Find a formula for the nth term.
Start by finding the first and second differences
Sequence: 5, 7, 11, 17, 25
First differences: 2, 4, 6, 8, ...
Second difference: 2, 2, 2, ...
Hence
a = 2 ÷ 2 = 1
Now write down an2 (just n2 in this case as a = 1) with the sequence underneath, and on the next line write the difference between an2 and the sequence
an2. : 1, 4, 9, 16, ...
sequence: 5, 7, 11, 17, ...
difference: 4, 3, 2, 1, ...
Work out the nth term of these differences to give you bn + c
bn + c = −n + 5
Add an2 and bn + c together to give you the nth term of the sequence
nth term = n2 − n + 5
(b)
Hence find the 20th term of the sequence.
Substitute n = 20 into n2 − n + 5
Substitute n = 20 into n2 − n + 5
(20)2 − 20 + 5 = 400 − 15
20th term = 385
